Numerical Infinities and Infinitesimals in Optimization (Emergence, Complexity and Computation, 43) PDF
Preview Numerical Infinities and Infinitesimals in Optimization (Emergence, Complexity and Computation, 43)
Emergence, Complexity and Computation ECC Yaroslav D. Sergeyev Renato De Leone Editors Numerical Infinities and Infinitesimals in Optimization Emergence, Complexity and Computation Volume 43 SeriesEditors IvanZelinka,TechnicalUniversityofOstrava,Ostrava,CzechRepublic AndrewAdamatzky,UniversityoftheWestofEngland,Bristol,UK GuanrongChen,CityUniversityofHongKong,HongKong,China EditorialBoard AjithAbraham,MirLabs,USA AnaLucia,UniversidadeFederaldoRioGrandedoSul,PortoAlegre,Rio GrandedoSul,Brazil JuanC.Burguillo,UniversityofVigo,Spain SergejCˇelikovský,AcademyofSciencesoftheCzechRepublic,Czech Republic MohammedChadli,UniversityofJulesVerne,France EmilioCorchado,UniversityofSalamanca,Spain DonaldDavendra,TechnicalUniversityofOstrava,CzechRepublic AndrewIlachinski,CenterforNavalAnalyses,USA JouniLampinen,UniversityofVaasa,Finland MartinMiddendorf,UniversityofLeipzig,Germany EdwardOtt,UniversityofMaryland,USA LinqiangPan,HuazhongUniversityofScienceandTechnology,Wuhan,China GheorghePa˘un,RomanianAcademy,Bucharest,Romania HendrikRichter,HTWKLeipzigUniversityofAppliedSciences,Germany JuanA.Rodriguez-Aguilar ,IIIA-CSIC,Spain OttoRössler,InstituteofPhysicalandTheoreticalChemistry,Tübingen, Germany YaroslavD.Sergeyev,UniversityofCalabria,Italy VaclavSnasel,TechnicalUniversityofOstrava,CzechRepublic IvoVondrák,TechnicalUniversityofOstrava,CzechRepublic HectorZenil,KarolinskaInstitute,Sweden The Emergence, Complexity and Computation (ECC) series publishes new developments,advancementsandselectedtopicsinthefieldsofcomplexity, computationandemergence.Theseriesfocusesonallaspectsofreality-based computation approaches from an interdisciplinary point of view especially from applied sciences, biology, physics, or chemistry. It presents new ideas andinterdisciplinaryinsightonthemutualintersectionofsubareasofcompu- tation,complexityandemergenceanditsimpactandlimitstoanycomputing basedonphysicallimits(thermodynamicandquantumlimits,Bremermann’s limit, Seth Lloyd limits…) as well as algorithmic limits (Gödel’s proof and its impact on calculation, algorithmic complexity, the Chaitin’s Omega numberandKolmogorovcomplexity,non-traditionalcalculationslikeTuring machineprocessanditsconsequences,…)andlimitationsarisinginartificial intelligence.Thetopicsare(butnotlimitedto)membranecomputing,DNA computing, immune computing, quantum computing, swarm computing, analogiccomputing,chaoscomputingandcomputingontheedgeofchaos, computational aspects of dynamics of complex systems (systems with self-organization, multiagent systems, cellular automata, artificial life,…), emergence of complex systems and its computational aspects, and agent based computation. The main aim of this series is to discuss the above mentioned topics from an interdisciplinary point of view and present new ideascomingfrommutualintersectionofclassicalaswellasmodernmethods ofcomputation.Withinthescopeoftheseriesaremonographs,lecturenotes, selectedcontributionsfromspecializedconferencesandworkshops,special contributionfrominternationalexperts. IndexedbyzbMATH. More information about this series at https://link.springer.com/bookseries/ 10624 · Yaroslav D. Sergeyev Renato De Leone Editors Numerical Infinities and Infinitesimals in Optimization Editors YaroslavD.Sergeyev RenatoDeLeone DipartimentodiIngegneria ScuoladiScienzeeTecnologie Informatica,Modellistica,Elettronicae UniversitàdegliStudidiCamerino Sistemistica Camerino,Italy UniversityofCalabria Rende,Italy LobachevskyUniversity NizhnyNovgorod,Russia ISSN2194-7287 ISSN2194-7295 (electronic) Emergence,ComplexityandComputation ISBN978-3-030-93641-9 ISBN978-3-030-93642-6 (eBook) https://doi.org/10.1007/978-3-030-93642-6 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringer NatureSwitzerlandAG2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher, whetherthewholeorpartofthematerialisconcerned,specificallytherightsoftranslation, reprinting,reuseofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinany otherphysicalway,andtransmissionorinformationstorageandretrieval,electronicadapta- tion, computer software, or by similar or dissimilar methodology now known or hereafter developed. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthis publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesare exemptfromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationin thisbookarebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublisher northeauthorsortheeditorsgiveawarranty,expressedorimplied,withrespecttothematerial containedhereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremains neutralwithregardtojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland The editors thank their families for their love, support, and really infinite patience during the years of preparation of this book on ➀ infinity. To Adriana and Ekaterina To Chiara, Dmitriy, and Robert Preface Thisbookisdedicatedtotwotopicsthat,uptotherecenttimes,werebelieved incompatible.Infact,infinitiesandinfinitesimalsforcenturieswereconsid- ered within highly theoretical research areas whereas optimization is a part of applied mathematics and deals with algorithms and codes implemented oncomputers.Thepossibilitytojointhesetwotopicsunderthesamecover isaconsequenceofthefactthatinfinitiesandinfinitesimalsconsideredhere are not traditional symbolic entities but numerical ones, namely, they can be represented on a new supercomputer patented in several countries and called the Infinity Computer. This computer is able to execute numerical (i.e.,floating-point)operationswithnumbersthatcanhavedifferentinfinite, finite, and infinitesimal parts represented in the positional numeral system withaninfinitebasecalledgrossoneexpressedbythenumeral➀.Thenumer- ical character of these infinite and infinitesimal numbers is one of the key differenceswithrespecttotraditionaltheoriesofinfinityandinfinitesimals. This novel point of view opens completely new horizons not only in pure mathematicsbutalsoincomputerscienceandappliedmathematics. Duringthelastyearstherewasanincreasinginterestinthisnewcomputa- tionalparadigm.In2013,2016,and2019therewereinternationalconferences “NUMTA—Numerical Computations: Theory and Algorithms” where the InfinityComputeranditsapplicationswereamongthemaintopics.Several special issues in leading scientific journals have been published. The last NUMTAconferencein2019hadmorethan200participantsfrom30coun- tries. Three special issues and proceedings in volumes 11973 and 11974 of the Springer book series “Lecture Notes in Computer Science” have been published. Theinterestoftheinternationalscientificcommunitytothisparadigmis also due to the fact that, for the first time, infinities and infinitesimals were vii viii Preface involvednotonlyinscientificfieldsrelatedtopuremathematicssuchasfoun- dationsofmathematics,logic,andphilosophybutalsoincomputerscience andappliedmathematics.Moreover,thereexistseveralsoftwaresimulatorsof theInfinityComputerthatareactivelyusedbyresearchersintheirworkwith numerical methods. Nowadays among the fields of application of the new computational paradigm we find: numerical differentiation and numerical solution of ordinary differential equations, game theory, probability theory, cellularautomata,Euclideanandhyperbolicgeometry,percolation,fractals, infiniteseriesandtheRiemannzetafunction,thefirstHilbertproblem,Turing machines,teaching,etc.AhugebulkofresultsusingtheInfinityComputer paradigm in optimization (linear, non-linear, global, and multi-criteria) is collectedinthisvolumeco-authoredby22leadingexpertsworkinginthese fields. Thus, the reader will have a unique opportunity to find in the same book the state-of-the-art knowledge and practices in optimization using the InfinityComputingmethodology. The first two chapters of the book written by the editors have an intro- ductorycharacteranddescribetheInfinityComputermethodology(Chapter “ANewComputationalParadigmUsingGrossone-BasedNumericalInfini- ties and Infinitesimals”, written by Yaroslav D. Sergeyev) and some of the most important results in unconstrained and constrained optimization (Chapter “Nonlinear Optimization: A Brief Overview”, written by Renato De Leone). Chapter “The Role of grossone in Nonlinear Programming andExactPenaltyMethods”,alsowrittenbyRenatoDeLeone,isdedicatedto exactpenaltymethodsforsolvingconstrainedoptimizationproblems.Using penaltyfunctions,theoriginalconstrainedoptimizationproblemcanbetrans- formed in an unconstrained one. The chapter shows how ➀ can be utilized in constructing exact continuously differentiable penalty functions for the caseofonlyequalityconstraints,thegeneralcaseofequalityandinequality constraints, and quadratics problems. It is also discussed how these new penaltyfunctionsallowonetorecovertheexactsolutionoftheunconstrained problemfromtheoptimalsolutionoftheunconstrainedproblem.Moreover, Lagrangian duals associated to the constraints can also be automatically obtainedthankstothe➀-basedpenaltyfunctions. Chapter “Krylov-Subspace Methods for Quadratic Hypersurfaces: aGrossone–basedPerspective”iswrittenbyGiovanniFasanowhostudiesthe roleof➀todealwithtworenownedKrylov-subspacemethodsforsymmetric (possiblyindefinite)linearsystems.First,theauthorexploresarelationship between the Conjugate Gradient method and the Lanczos process, along with their role of yielding tridiagonal matrices which retain large infor- mation on the original linear system matrix. Then, he shows that coupling Preface ix ➀ with the Conjugate Gradient method provides clear theoretical improve- ments. Furthermore, reformulating the iteration of this algorithm using this ➀-based framework allows the author to encompass also a certain number of Krylov-subspace methods relying on conjugacy among vectors. The last generalizationremarkablyjustifiestheuseofa➀-basedreformulationofthe Conjugate Gradient method to solve also indefinite linear systems. Finally, pairing this method with the algebra of grossone easily provides relevant geometricpropertiesofquadratichypersurfaces. Marco Cococcioni, Alessandro Cudazzo, Massimo Pappalardo, and Yaroslav D. Sergeyev show in Chapter “Multi-objective Lexicographic Mixed-IntegerLinearProgramming:AnInfinityComputerApproach”how a lexicographic multi-objective linear programming problem can be trans- formed into an equivalent single-objective problem by using the grossone methodology. Then, the authors provide a simplex-like algorithm, called GrossSimplex, able to solve the original problem using a single run of the algorithm. In the second part of the chapter, the authors tackle the mixed- integerlexicographicmulti-objectivelinearprogrammingproblemandsolve it in an exact way, by using a ➀-version of the Branch-and-Bound scheme. Afterprovingthetheoreticalcorrectnessoftheassociatedpruningrulesand terminatingconditions,theypresentafewexperimentalresultsexecutedon anInfinityComputersimulator. In Chapter “The Use of Infinities and Infinitesimals for Sparse Classi- fication Problems”, Renato De Leone, Nadaniela Egidi, and Lorella Fatone discusstheuseofgrossoneindeterminingsparsesolutionsforspecialclasses of optimization problems. In fact, in various optimization and regression problems, and in solving overdetermined systems of linear equations it is oftennecessarytodetermineasparsesolution,thatisasolutionwithasmany aspossiblezerocomponents.Theauthorsshowhowcontinuouslydifferen- tiable concave approximations of the pseudoâe“norm can be constructed using ➀ and discuss properties of some new approximations. Finally, the authors present some applications in elastic net regularization and Sparse SupportVectorMachine. Chapter “The Grossone-Based Diagonal Bundle Method” written by Manlio Gaudioso, Giovanni Giallombardo, and Marat S. Mukhametzhanov discussesafruitfulimpactoftheInfinityComputingparadigmonthepractical solutionofconvexnonsmoothoptimizationproblems.Theauthorsconsider aclassofmethodsbasedonavariablemetricapproach:theuseoftheInfinity Computingtechniquesallowsthemtonumericallydealwithquantitieswhich cantakearbitrarilysmallorlargevalues,asaconsequenceofnonsmoothness. Inparticular,bychoosingadiagonalmatrixwithpositiveentriesasametric, the authors modify the well-known Diagonal Bundle algorithm by means x Preface of matrix updates based on the Infinity Computing paradigm and provide computationalresultsobtainedonasetofbenchmarktestproblems. Chapter “On the Use of Grossone Methodology for Handling Priorities in Multi-objective Evolutionary Optimization” written by Leonardo Lai, Lorenzo Fiaschi, Marco Cococcioni, and Kalyanmoy Deb describes a new class of problems, called mixed Pareto-lexicographic multi-objective opti- mizationproblems,asuitablemodelforscenarioswheresomeobjectiveshave priorityoversomeothers.Tworelevantsubclassesofthisproblemareconsid- ered:prioritychainsandprioritylevels.ItisshownthattheInfinityComputing methodologyallowsonetohandleprioritiesefficiently.Itisremarkablethat this technique can be easily embedded in most of the existing evolutionary algorithms,withoutalteringtheircorelogic.Threealgorithmsaredescribed and tested on benchmark problems, including some real-world problems. The experiments show that the algorithms using the Infinity Computing methodologyareabletoproducemoresolutionsandofhigherquality. Chapter “Exact Numerical Differentiation on the Infinity Computer and Applications in Global Optimization” written by Maria Chiara Nasso and Yaroslav D. Sergeyev shows how exact numerical differentiation can be executed on the Infinity Computer and efficiently applied in Lipschitz globaloptimizationforcomputingderivatives.Optimizationalgorithmsusing smooth piece-wise quadratic support functions to approximate the global minimumarealsodiscussedandtheirconvergenceconditionsareprovided. It is shown that all the methods can be implemented both in the tradi- tional floating-point arithmetic and in the novel Infinity Computing frame- work.Numericalexperimentsconfirmthatmethodsusinganalyticderivatives and derivatives computed numerically on the Infinity Computer exhibit the identicalbehavior. Chapter “Comparing Linear and Spherical Separation Using Grossone-Based Numerical Infinities in Classification Problems” authored by Annabella Astorino and Antonio Fuduli investigates the role played by the linear and spherical separations in binary supervised learning and in Multiple Instance Learning (MIL), in connection with the use of the grossone-based numerical infinities. While in classical binary supervised learning the objective is to separate two sets of samples, a binary MIL problemconsistsinseparatingtwodifferenttypeofsets(positiveandnega- tive), each of them constituted by a finite number of samples. The authors focusonthepossibilitytoconstructbinarysphericalclassifierscharacterized by an infinitely far center adopting the Infinity Computing methodology. They show that this approach allows them to obtain a good performance in terms of average testing correctness and to manage very easily numerical computationswithoutanytuningofthe“bigM”parameter.
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